Ultrasonic characterization of functionally graded materials using a continuously graded model and spectral inversion

Quantitative ultrasonic characterization of functionally graded materials (FGMs) remains challenging because conventional discrete layered models (DLMs) approximate continuous gradients as sharp internal interfaces, which can introduce non-physical spectral features. In this study, a continuously graded model (CGM) is developed to describe ultrasonic wave propagation in FGMs and is combined with particle swarm optimization (PSO) to estimate layer thicknesses from the ultrasonic reflection coefficient spectrum (URCS). The proposed CGM captures the smooth variation of acoustic impedance within graded regions and reduces the artificial interface-induced reflections associated with discrete approximations. Sensitivity analysis indicates that resonance troughs in the URCS provide informative features for thickness inversion. Experimental validation on a three-layer W-Cu/SiC FGM shows that the proposed framework can estimate layer thicknesses with relative errors of less than 6 %, even when the initial guesses deviate significantly from the reference values. These results suggest that the CGM-PSO framework provides a physically consistent and promising approach to ultrasonic characterization of continuously graded materials.